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To get started you can use oeis.org to investigate this. For instance, from plugging in the numerators corresponding to some of your data it seems that $$\#P_n^{(2)}(1)=n(n-1)$$ ("the oblong numbers") and $$\#P_n^{(6)}(1)=n(n-1)(n-2)^4$$ ("the number of $n$-colorings of the Triangle Graph of order 3").